# Linear representation of the free metabelian / 2-step nilpotent profinite groups on 2 generators

Let G be the free profinite group on 2 generators, $$A=G/[G,[G,G]],B=G/[[G,G],[G,G]]$$, then what is the structure of the groups $$A$$ and $$B$$?

I heard that $$A$$ is isomorphic to the group of such ($$3\times 3$$ below) matrices with entries in $$\hat{\mathbb{Z}}$$, is this right and why? $$\begin{pmatrix} 1 & * & *\\ 0 & 1 & *\\ 0 & 0 & 1 \end{pmatrix}$$

The group $$B$$, the free pro-metabelian group, has the following description, due to Jorge Almeida. I’ll do it for an arbitrary finite set $$|X|$$ of cadinality at least $$2$$. Consider $$\widehat{\mathbb Z}^X$$, the free pro-abelian group on $$X$$. Then we can consider the edge set of its Cayley graph $$E=\widehat{\mathbb Z}^X\times X$$, which is a profinite space with the product topology. Let $$H$$ be the free pro-abelian group on the profinite space $$E$$. Then $$\widehat{\mathbb Z}^X$$ acts continuously on $$E$$ via the usual action on its Cayley graph, i.e., via left multiplication in the first coordinate and this extends to a continuous action on $$H$$ by automorphisms. Form the semidirect product $$H\rtimes \widehat{\mathbb Z}^X$$. Then your group $$B$$ embeds in $$H\rtimes \widehat{Z}^X$$ in the following way. Send $$x\in X$$ to the pair $$((1,x),x)$$ where $$(1,x)$$ should be thought of as the edge from $$1$$ to $$x$$ labeled by $$x$$ in the Cayley graph and the second $$x$$ is the corresponding generator of $$\widehat{\mathbb Z}^X$$. This extends to an embedding of $$G$$.
Your question about $$A$$ boils down to whether the $$3\times 3$$ Heisenberg group has the congruence subgroup property, which I leave to more knowledgeable people than I.
• Yes it's very easy to show by hand that the $3\times 3$-Heisenberg group has the congruence subgroup property. More generally this holds in general, in the sense that for every unipotent $\mathbf{Q}$-subgroup $U$ of $\mathrm{GL}_d$, every finite index subgroup of $\mathrm{GL}_d(\mathbf{Z})\cap G(\mathbf{Q})$ has the congruence subgroup property. – YCor Dec 2 '18 at 17:39